FUNDAMENTAL THEOREM OF ALGEBRA


Here we can see the complex plane at different scales: zoomed in and zoomed out.
We are visualizing inputs of constant modulus (or magnititude) A in yellow, and outputs of the polynomial function in green. We can follow a specific input and output by controlling the theta parameter which does a single full circle through the input values.

For large modulus A, how many times does the specific output value circle the origin as theta increases from 0 to 2pi?
When A=0, what is the output of the polynomial?
Now consider what has to happen as A decreases from larger values to zero, how many times must it pass through the origin?
Note that every time the output passes through the origin we have found a root of our polynomial, so how many roots must it have?